Given a compact group $G\leq \operatorname{O}(d)$ of linear isometries on $\mathbb R^d$, equip its quotient $\mathbb R^d/G$ with the canonical orbital metric.

I am curious about the following. Is there an example of $G$ for which there is a minimal geodesic $\gamma\colon [0,t_0]\to \mathbb R^d/G$ such that $t_0$ is a cut-point of $\gamma$, yet $\gamma$ is the unique minimal geodesic joining $\gamma(0)$ to $\gamma(t_0)$?

Here, a geodesic in $\mathbb R^d/G$ means a projection of a horizontal geodesic in $\mathbb R^d$, and the cut-point $t_0$ means that $\gamma$ fails to be minimizing shortly beyond $t_0$.

This might have a relationship with the plethora of horizontal conjugate points ensured to exist near non-orbifold points in $\mathbb R^d/G$ (see this paper). However, I am not sure whether linear isometry quotients $\mathbb R^d/G$ are so nice that horizontal conjugate points can never be connected by a unique minimal geodesic.